No universal group in a cardinal

by Shelah. [Sh:1029]
Forum Math, 2016
For many classes of models there are universal members in any cardinal lambda which ``essentially satisfied GCH'', i.e. lambda = 2^{<= lambda} . But if the class is ``complicated enough'', e.g. the class of linear orders, we know that if lambda is ``regular and not so close to satisfying GCH'' then there is no universal member. Here we find new sufficient conditions (which we call the olive property),

not covered by earlier cases (i.e. fail the so-called SOP_4). The advantage of those conditions is witnessed by proving that the class of groups satisfies one of those conditions.

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